Check the author field in the right column. Make sure it it set for "two fields," not single. This puts last name first.

To change, click the little white space that is between the name and the - + signs.

If this fixes your problem, you will need to change each item that has single field selected.

If it doesn't fix your problem, please write back!

and as you see in the example it's only for one of the authors however the second author also have a first name letter in the database

It's not style-specific. I'm experiencing this with all (Author-Date) styles

I got your email and could not replicate the issue, I will play with a bit more and see if I can. In your email you mentioned

The same reference can be correct and wrong in the same word-document

With that noted, and the fact that they worked fine for me we can rule out an issue with the styles and the way the names are formated in your collection.

When you hit the refresh button do they switch back to normal?

The problem happens to me if the author uses her first name in a book chapter, but her initials in an article, or her first name and middle initial. If there's any variance, the software works extra hard to distinguish the variants as different authors, I think.

T. S. Eliot and Thomas Stearns Eliot would each get cited in the text as different people. The only way I can make it stop is to edit the entries to be consistent, which is not what the author or editor intended.

Zotero, and for that matter any other bibliographic software I can think of, simply have no way to know that T. S. Eliot is the same person as Thomas Sterns Eliot as opposed to a different person named Trevor Stanley Eliot.

For the foreseeable future you will need to make sure that authors are referred to by the same name for Zotero to know that they are the same person.